A110104 a(n) is the number of coverings of 1..n by cyclic words of length 3n, such that each value from 1 to n appears precisely twice. That is, the union of all the letters in all of the words of a given covering is the multiset {1,1,2,2,...,n,n}. No repeats of words are allowed in a given covering.

1, 4, 3760, 23504320, 567399078400, 37518268781593600, 5543744611870143078400, 1599334510537656091623424000, 818296434784062385011283591168000

Offset

0,2

Comments

P-recursive.

Links

Table of n, a(n) for n=0..8.

Formula

Differential equation satisfied by egf: sum a(n)t^3n/(3n!) {F(0) = 1, (-2+4*t^6+16*t^3)*(d/dt)F(t) + 4*t^4*(d^2/dt^2)F(t) + t^2*(4+12*t^3+t^6)*F(t)}.

Recurrence: {a(0) = 1, (40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8)*a(n) +(508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2)*a(n+1) + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4)*a(n+2) - 2*a(n+3), a(1) = 4, a(2) = 3760}.

a(n) ~ 2^n * 3^(4*n + 1/2) * n^(4*n) / exp(4*n). - Vaclav Kotesovec, Oct 24 2023

Example

a(1)=4: {123, 132} {112, 233} {113, 322} {133, 122}

Mathematica

RecurrenceTable[{(40320 + 328752*n + 1816668*n^3 + 1102248*n^5 + 398034*n^6 + 1818369*n^4 + 1063116*n^2 + 78732*n^7 + 6561*n^8) * a[n] + (508608*n + 161280 + 453600*n^3 + 34992*n^5 + 2916*n^6 + 173340*n^4 + 661104*n^2) * a[n + 1] + (12320 + 19980*n + 12096*n^2 + 3240*n^3 + 324*n^4) * a[n + 2] - 2*a[n + 3] == 0, a[0] == 1, a[1] == 4, a[2] == 3760}, a, {n, 0, 15}] (* Vaclav Kotesovec, Oct 24 2023 *)

Crossrefs

Cf. A052502, A110105, A110106, A108242.

Sequence in context: A114498 A069120 A221191 * A024061 A013830 A320860

Adjacent sequences: A110101 A110102 A110103 * A110105 A110106 A110107

Keyword

easy,nonn

Author

Marni Mishna, Jul 11 2005

Status

approved